Extending Brauer's height zero conjecture to blocks with nonabelian defect groups. (Q2929653)

Let \(p\) be a prime, and let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(D\). The authors denote by \(mh(B)\) the minimal nonzero height of an irreducible character in \(B\); if all irreducible characters in \(B\) have height zero then \(mh(B)\) is interpreted as \(\infty\). Moreover, \(mh(D)\) denotes \(mh(b)\) where \(b\) is the only \(p\)-block of \(D\). The authors conjecture that \(mh(B)=mh(D)\). This generalizes Brauer's height zero conjecture which asserts that \(D\) is abelian if and only if all irreducible characters in \(B\) have height zero. The authors give strong evidence for the inequality \(mh(B)\geq mh(D)\); for example, this follows from Dade's projective conjecture. In particular, it is true for \(p\)-solvable groups. The authors also present some partial results concerning the other inequality.